A hostage and two robbers were killed in a shootout with California police that followed a violent bank heist.

Three gunmen raided a Bank of the West branch in northern Stockton, about 50 miles south of Sacramento, early in the afternoon.

Armed with AK-47 rifles, the trio took three women hostage and left the premises.

Police that responded an emergency call and arrived at the scene gave chase to the hostage takers, who made off in a blue Ford Explorer into the nearby city of Lodi and then back into Stockton.

"During the chase, the suspects continually shot at officers with AK-47 style rifles," the Stockton police department said. Cars and houses along their route were hit by gunfire.

"The suspects had mass amounts of ammunition, even with some firearm magazines taped or strapped to their bodies," Stockton police said. "The gunfire on our officers was relentless."

Police said that shortly into the one-hour pursuit, one of the hostages was shot and thrown out of the fleeing vehicle. The injured woman was taken to a hospital and is expected to survive, authorities said.

Similarly, a second hostage was later thrown from or managed to jump off the car, which was travelling at more than 50 mph, police said. She is also expected to recover from her injuries.

The Ford Explorer finally came to a standstill after being disabled by security forces and a further gun battle ensued.

Two suspects were shot dead. The third hostage - identified as Misty Holt-Singh, a mother to a 12-year-old girl - was left fatally wounded on the back seat.

The third robber, named as Stockton resident Jaime Ramos, 19, has been arrested. He is facing homicide, kidnapping, robbery, and attempted murder charges.

Police said that Ramos's deceased accomplices "were documented gang members and residents of Stockton" aged 30 and 27.

The final part of the chase was captured on camera by local residents in this video posted online.